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Harald van Brummelen (TU/e)
| Speaker: |
Prof.dr.ir. Harald van
Brummelen (TU/e, Dept. of Mechanical Engineering)
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| Date: |
Wednesday October 21, 2009
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| Title: |
Goal-oriented error
estimation and adaptivity for multiscale flow problems
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Abstract
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Numerical simulations of flow
problems that display a wide range of scales generally require vast
computational resources. Typical examples of such multiscale flow
problems are fluid-structure interaction and transitional
molecular/continuum flows. For fluid-structure interaction, the
multiscale behavior emanates from the distinct length and time scales
in the fluid and structure subsystems.
In addition, the interface between the fluid and the structure
represents a free boundary, which generally introduces additional
scales. For transitional molecular/continuum flows, the ratio between
the molecular mean free path and a typical length scale of observation
is such that the flow is too far from equilibrium to apply standard
continuum models such as the Euler or Navier-Stokes equations, while on
the other hand the overall number of molecules in the system is too
large to apply a molecular-dynamics model. Similarly, the flow can be
in equilibrium in large parts of the domain, but may locally display
significant non-equilibrium effects, e.g., due to interactions at
material interfaces which are incompatible with the equilibrium
distribution.
In many applications, interest is restricted to a single goal
functional. For instance, in fluid-structure interaction, one may be
interested in the energy which is transferred from the fluid to the
structure during a certain time interval. Such an energy-transfer
functional fully characterizes the stability of the fluid-structure
system. In heat-transfer problems pertaining to transitional
molecular/continuum flows, interest is generally restricted to the heat
flux across a certain part of the boundary.
A crucial notion concerns the fact that the restricted interest to a
particular goal functional can be exploited to reduce the complexity of
the system. This notion is formalized by goal-oriented a-posteriori
error estimation and optimal adaptive-refinement methodologies. By
means of the solution of an appropriate dual problem, the contribution
of local errors in the solution to the error in the goal functional can
be established. Only the regions that have a pronounced influence on
the error in the goal functional need to be refined. Such an approach
can be used for both discretization adaptivity and model adaptivity. In
the first form of adaptivity, the computational mesh (h) or order of
approximation (p) is locally adapted to reduce the error. In the second
form of adaptivity, the underlying model is locally adapted, e.g., by
locally replacing a continuum model by a molecular model.
In this presentation, I will give an overview of goal-oriented error
estimation and adaptive-refinement techniques, and I will discuss the
application to fluid-structure-interaction problems and to
molecular/continuum flows.
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